In a class of $9$, there are $5$ students who are secretly robots. If the teacher chooses $3$ students, what is the probability that none of the three of them are secretly robots?
Solution: We can think about this problem as the probability of $3$ events happening. The first event is the teacher choosing one student who is not secretly a robot. The second event is the teacher choosing another student who is not secretly a robot, given that the teacher already chose someone who is not secretly a robot, and so on. The probabilty that the teacher will choose someone who is not secretly a robot is the number of students who are not secretly robots divided by the total number of students: $\dfrac{4} {9}$ Once the teacher's chosen one student, there are only $8$ left. There's also one fewer student who is not secretly a robot, since the teacher isn't going to pick the same student twice. So, the probability that the teacher picks a second student who also is not secretly a robot is $\dfrac{3} {8}$ The probability of the teacher picking two students who are not secretly robots must then be $\dfrac{4} {9} \cdot \dfrac{3} {8}$ We can continue using the same logic for the rest of the students the teacher picks. So, the probability of the teacher picking $3$ students such that none of them are secretly robots is $\dfrac{4}{9}\cdot\dfrac{3}{8}\cdot\dfrac{2}{7} = \dfrac{1}{21}$